Measurement-based quantum computation in finite one-dimensional systems string order implies com甭瑡椀漭湡l power

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Measurement-based quantum computation in ﬁnite

one-dimensional systems: string order implies

computational power

Robert Raussendorf1,2, Wang Yang3, and Arnab Adhikary4,2

1Leibniz University Hannover, Hannover, Germany

2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, Canada

3School of Physics, Nankai University, Tianjin, China

4Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada

We present a new framework for assessing the power of measurement-based

quantum computation (MBQC) on short-range entangled symmetric resource

states, in spatial dimension one. It requires fewer assumptions than previously

known. The formalism can handle ﬁnitely extended systems (as opposed to

the thermodynamic limit), and does not require translation-invariance. Fur-

ther, we strengthen the connection between MBQC computational power and

string order. Namely, we establish that whenever a suitable set of string order

parameters is non-zero, a corresponding set of unitary gates can be realized

with ﬁdelity arbitrarily close to unity.

1 Introduction

Resource states for measurement-based quantum computation (MBQC) [1] are known to

be rare in Hilbert space [2]. But symmetry adds a twist to this picture. When symmetries

are present, in the thermodynamic limit, short-range entangled quantum states group

into so-called computational phases of quantum matter [3–8]. From a condensed matter

perspective, these phases are symmetry protected topologically (SPT) ordered [9–13].

From the perspective of quantum computation, these phases are warehouses full of MBQC

resource states. Any quantum state in a given SPT phase can be used to realize quantum

computations, and, moreover, the same quantum computations. The power of MBQC

across SPT phases is uniform [15–22].

The phenomenology of MBQC becomes richer with increasing spatial dimension of the

resource states: one dimension (1D) is mostly a test bed for computational methods, 2D

reaches quantum computational universality [1,23,24], and 3D combines universality with

fault-tolerance [14]. This increase of computational power with dimension is matched in

computational phases. The ﬁrst such phases were identiﬁed in 1D [15–18], capable of pro-

cessing a bounded number of logical qubits. In 2D, examples of universal computational

phases are known [19–22]. In 3D, the fault-tolerance capability of cluster states has been

related to SPT order with 1-form symmetry [25]. As the phenomenology ﬂourishes with

increasing dimension, our understanding diminishes: In spatial dimension one, a classiﬁ-

cation scheme for computational phases exists [16–18]; and furthermore a gauge principle

underlying MBQC has been identiﬁed [27]. In higher dimensions we have several examples

for computational phases, but no classiﬁcation.

Accepted in Quantum 2023-11-24, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.05089v4 [quant-ph] 25 Dec 2023

For the reasons just outlined, most current research on the subject of computational

phases of quantum matter focuses on higher dimensions. Nonetheless, in the present

paper we return to the one-dimensional case, to devise a more versatile formalism for the

discussion of MBQC in the presence of symmetry. We do this with the intention of later

applying it to 2D and 3D, and beyond that, to identify a unifying framework in which

the subjects of foundational interest in MBQC—contextuality, symmetry, temporal order,

topological fault-tolerance and gauge principle—can all be discussed. At the beginning of

our exploration stands the question: How is MBQC computational power on symmetric

states aﬀected if we transition from inﬁnite to ﬁnite systems?

The question is well-motivated: quantum computation is about eﬃciency, hence re-

source counting. The ﬁnite size of an MBQC resource state is thus an essential property.

Yet our main interest is conceptual: if we turn to ﬁnite systems, the notion of ‘symmetry

protected phase’ dissolves. But then, what happens to the cohomological classiﬁcation of

resource states, hence MBQC schemes?

We are prompted to adopt a novel perspective. Namely, in the discussion of compu-

tational phases of quantum matter to date [7,8,15–22], the resource state is the primary

object, the object to classify. The measurement procedure that extracts computational

power is almost an afterthought. Now we turn this picture on its head. Phases—symmetry-

protected, computational, or otherwise—are not deﬁned in ﬁnite systems. This is a priori

a detriment, for the classiﬁcation of SPT order in terms of group cohomology [9–13] hinges

on it. Group cohomology is also the basis for the “SPT-to-MBQC meat grinder” [16,17],

which converts cohomological data into MBQC schemes.

As we show in this paper, in the new situation of ﬁnite system size, the measurement

procedure takes over as the primary object, the object suited to classiﬁcation. Projective

representations, and their cohomological classiﬁcation, reappear in it. The resource states,

in turn, become the accessory in the formalism. They have to be short-range entangled,

symmetric, and possess string order matching the symmetry. And that’s all there’s to say

about them. A ﬁrst implication of this reversal is that a characterization of MBQC on

symmetric resource states in terms of group cohomology can be retained for ﬁnite systems.

Advantages of the new formalism—ranging from the conceptual to the more practical—

are as follows. (I) We strengthen the connection between string order and computational

power of MBQC in one dimension. Namely we show that, as long as string order [28–30]

is present, however weak, arbitrarily accurate non-trivial computation is possible. (II) We

align the MBQC notion of locality (site local) with the SPT notion of local (previously

block-local), and (III) We no longer require translation-invariance of the resource state.

The remainder of this paper is organized as follows. In Section 2we describe the

above-listed advances in greater detail. In Section 3we deﬁne our setting, and introduce

the four examples through which we will subsequently illustrate our result, namely the

cluster chain, the Kitaev-Gamma chain, a spin chain relating to the output of a Cliﬀord

quantum cellular automaton (QCA), and the Ising chain with transverse magnetic ﬁeld.

In Section 5we state and prove our main result, Theorem 1. It says that multi-particle

quantum states can be used as resources for measurement based quantum computation if

they (a) are invariant under a suitable group of symmetries, (b) are short-range entangled,

and (c) have non-vanishing string order parameters of a form matching the symmetries.

We apply the theorem to the examples introduced in the previous section. Section 6is

about block locality vs. site locality. Here we treat the cluster chain and the QCA chain

in a reﬁned fashion, leading to blocks of size one. In Section 7we discuss the relation

between string order parameters and the computational order parameters deﬁned in [16].

In Section 8we relate string order to quantum contextuality. Section 9is the conclusion.

Accepted in Quantum 2023-11-24, click title to verify. Published under CC-BY 4.0. 2

2 Advances of the new formalism

We now explain the advances made by the new formalism.

(1) Computational order equals string order: The relevance of string operators for the

functioning of MBQC was ﬁrst recognized in [3,4]. In [4], quantum correlations describing

the ﬁdelity of gate simulations in MBQC were expressed in terms of string operators.

In [3], it was shown for ground states of the transverse ﬁeld cluster model, the gate ﬁdelity

is bounded from below by a constant.

Here, we strengthen the above connection. Namely we show that whenever the string

order parameters are non-zero, quantum gates can be realized in MBQC with ﬁdelity

arbitrarily close to unity. The higher the ﬁdelity targeted, the larger the section of resource

state consumed in the implementation of the gate.

In prior analysis of MBQC on resources states taken from SPT phases [16,17], in the

framework of MPS, a computational order parameter νwas identiﬁed that governs the

operational overheads of MBQC. It was shown in [16] how to extract this order parameter

from the MPS tensor representing the resource quantum state, but no physical interpre-

tation for it had been found. We now realize that the computational order parameter ν

and the string order parameter are the same.

(2) Block size: In the discussion of SPT and MBQC by the MPS formalism, neigh-

bouring spins are grouped into blocks [7,15–21], such that the action of the symmetry

group on each block is faithful. The block thereby becomes the natural local unit for the

formalism.

In all cases so far considered, the blocks comprise more than a single spin, and this

leads to a mismatch from the perspective of MBQC phenomenology. Namely, in standard

MBQC, the local unit is a single spin. The measurements driving an MBQC are supposed

to be site-local, not just block local. There is thus a gap between the MPS formalism

and the phenomenology of interest. In the prior discussions of 1D, the block size is only

2; a gap that was deemed minor. In 2D, however, the block size increases with system

size, leading to a very weak result about computational phases of quantum matter if left

unaddressed. Therefore, in [19–21], supplemental arguments have been put on top of the

basic formalism to reach block size one.

The present formalism doesn’t require faithfulness of the representations involved, and

can therefore handle blocks of any size down to size one. The physically motivated single-

site locality of MBQC can be matched by the present formalism in its very algebraic

structure, without the need for add-on arguments.

(3) Translation invariance: The prior formalism [7,15–21] requires translation invari-

ance whereas the present formalism doesn’t. Translation invariance is tied to the ther-

modynamic limit: no ﬁnite chain is translation-invariant. Therefore, getting rid of the

constraint of translation invariance is a precondition for discussing ﬁnite systems.

The present formalism achieves this, and in fact permits much greater ﬂexibility than

merely permitting the existence of boundaries. For example, the value of the string order

parameter may vary with the location of its end point in any fashion.

3 The setting

In this section we deﬁne our setting of short-range entangled symmetric states, and intro-

duce the examples that we will subsequently use to illustrate our main theorem.

Accepted in Quantum 2023-11-24, click title to verify. Published under CC-BY 4.0. 3

3.1 Symmetric short-range entangled states

As our fundamental notion of “short-range entangled”, we use that of short-range, bounded

depth quantum circuits applied to a product state. Two quantum states are considered

equivalent under a given symmetry Gif they can be related by a G-symmetric such circuit.

This is an operationally well-motivated notion in the context of quantum computation.

We consider quantum states |Φ⟩on open chains of spin 1/2 particles. The support of

the states |Φ⟩is grouped into nblocks in the bulk, plus a block 0on the left boundary

and a block n+ 1 on the right boundary. Graphically,

0 1 2 n n+1

block #

left boundary right boundary

bulk

The states |Φ⟩are short-range entangled and G-symmetric.

Symmetry. The symmetry group Gdiscussed in this paper is of the form G= (Z2)m.

It acts via a linear representation Uon |Φ⟩,

U(g)|Φ⟩= (−1)χ(g)|Φ⟩, χ(g)∈Z2,∀g∈G. (1)

Entanglement structure. The resource states |Φ⟩we consider are all of the form

|Φ⟩=WΦ(|+⟩|+⟩..|+⟩).(2)

Therein, WΦis a bounded-depth circuit composed of bounded-range gates. Symmetric

such states can arbitrarily closely approximate all ground states in SPT phases [34].

We quantify the short-range entangling nature of WΦas follows. We deﬁne two subsets

of particle block labels on the line,

{≤ k}:= {0,1,2, .., k},{> k}:= {k+ 1, k + 2, .., n + 1}.

The short-range nature of WΦis speciﬁed by an entanglement range ∆. Denoting by

supp(A)the support of a linear operator Aon the line segment {1, .., n}, we make the

following deﬁnition.

Deﬁnition 1 The entanglement range ∆of a quantum circuit WΦacting on the spin chain

{0, .., n + 1}is the smallest integer ∆≥0such that, for all k= 0, .., n + 1, it holds that

supp(W†

ΦAWΦ)⊂ {≤ (k+ ∆)},∀A|supp(A)⊂ {≤ k},

supp(W†

ΦAWΦ)⊂ {>(k−∆)},∀A|supp(A)⊂ {> k}.(3)

The short-range entanglement in resource states |Φ⟩enters MBQC through the following

lemma.

Lemma 1 Consider a short-range entangled state |Φ⟩=WΦ|+⟩|+⟩..|+⟩, where the circuit

WΦhas an entanglement range ∆. Be Aand Btwo linear operators, with their support

contained in {≤ (k−∆)}and {>(k+ ∆)}, respectively, for any k= ∆, .., n+1 −∆. Then

it holds that

⟨Φ|AB|Φ⟩=⟨Φ|A|Φ⟩⟨Φ|B|Φ⟩.(4)

Accepted in Quantum 2023-11-24, click title to verify. Published under CC-BY 4.0. 4

Proof of Lemma 1.We deﬁne L:= {≤ k},R:= {> k}, and write the product state

to which the short-range circuit WΦis applied as |+⟩LR := |+⟩L⊗ |+⟩R, with |+⟩L=

|+⟩0⊗.. ⊗ |+⟩kand |+⟩R=|+⟩k+1 ⊗.. ⊗ |+⟩n+1, with all |+⟩ireference states on blocks i,

respectively. Only locality between the left half Land the right half Rof the chain, split

between blocks kand k+ 1, matters.

We observe that, with the assumptions of the Lemma and Eq. (3) it holds that

supp(W†

ΦAWΦ)⊆ L and supp(W†

ΦBWΦ)⊆ R; hence

W†

ΦAWΦ=W†

ΦAWΦ|L⊗IR, W †

ΦBWΦ=IL⊗W†

ΦBWΦ|R.(5)

We then have

⟨Φ|AB|Φ⟩=L⟨+| ⊗R⟨+|W†

ΦABWΦ|+⟩L⊗ |+⟩R

=L⟨+| ⊗R⟨+|W†

ΦAWΦW†

ΦBWΦ|+⟩L⊗ |+⟩R

=L⟨+|W†

ΦAWΦL|+⟩L R⟨+|W†

ΦBWΦR|+⟩R

=LR⟨+|W†

ΦAWΦ|+⟩LR LR⟨+|W†

ΦBWΦ|+⟩LR

=⟨Φ|A|Φ⟩⟨Φ|B|Φ⟩.

Therein, in the third line we have used Eq. (5). □

3.2 The role of Hamiltonians in our setting

A comment about the role of Hamiltonians and their ground states in measurement based

quantum computation is now in order. From a fundamental point of view, MBQC has

nothing to do with Hamiltonians at all; it is only about states and measurements. Yet,

all examples in this paper consider ground states of Hamiltonians; see Section 3.3 below.

Here we explain this dichotomy.

First, Hamiltonians do ﬁnd a role to play in MBQC, in the following way. It was

observed in [2] that, when sampled uniformly from Hilbert space, computationally use-

ful resource states are extremely rare. This prompted the question: How frequent are

computational resources among quantum states that naturally occur? A common no-

tion of ‘naturally occurring’ is ground states of simple Hamiltonians. In this regard it

has been established, for example, that AKLT states in dimension two are universal for

MBQC [23,24].

The idea of ground states as computational resources fully came into its own with

the discovery of computational phases of quantum matter [3–7], when it was understood

that entire symmetry protected topological phases have computational power [15–18] and

can even be universal [19–21]. A counterpoint to the above scarcity of resource states

argument [2] is thereby made: in the presence of symmetry, computational resources are

no longer rare. The ground state manifold splits into extended phases, some of which

have computational power and others don’t. Computational phases of quantum matter

represent the strong case for invoking Hamiltonians in the discussion of MBQC.

In the present paper, we consider ﬁnite systems. The notion of ‘phase’ does therefore

no longer apply; and with it disappears the most enticing motivation for considering

Hamiltonians. However, the earlier motivation remains: Ground states model naturally

occurring states—this applies to ﬁnite systems just as well as to inﬁnite ones. There’s still

a case for invoking Hamiltonians.

A shift occurs with the formal criterion for ‘short-range entangled’ we impose, Eq. (3).

It is based on bounded-depth quantum circuits composed of short-range gates. The man-

ifold of quantum states described in this fashion has an operational motivation in its own

Accepted in Quantum 2023-11-24, click title to verify. Published under CC-BY 4.0. 5

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Measurement-basedquantumcomputationinfiniteone-dimensionalsystems:stringorderimpliescomputationalpowerRobertRaussendorf1,2,WangYang3,andArnabAdhikary4,21LeibnizUniversityHannover,Hannover,Germany2StewartBlussonQuantumMatterInstitute,UniversityofBritishColumbia,Vancouver,Canada3SchoolofPhysics,Nankai...

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Measurement-based quantum computation in finite one-dimensional systems string order implies com甭瑡椀漭湡l power

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